As announced in the previous section, the proof of Theorem REF  builds upon LDP for uncontrolled, interacting (forward) particle systems.
In order to exploit such results, we will introduce a forward-backward particle system that is "similar" to (but considerably more tractable than) (REF ).
Using this auxiliary particle system and well-known decoupling techniques from the theory of forward-backward SDEs allow to construct an uncontrolled forward particle system for which LDP results are well-known.
The last step of the proof is to show that the auxiliary particle system for which the LDP is known in the literature is "close enough" to our original particle system.
This is achieved using a priori estimations for FBSDEs.
Here, close enough should be understood in the following sense put forth in [1]}:
